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TIMEWITHOUT END: PHYSICS AND BIOLOGY IN ANOPENUNIVERSE

(*)

Freeman J. Dyson

Institute for Advanced Studies, Princeton New Jersey 08540

Reviews of Modern Physics, Vol. 51, No. 3, July 1979 (c) 1979 American Physical Society

Quantitative estimates are derived for three classes of phenomena that may occur in anopen cosmological model of Friedmann type. (1) Normal physical processes taking placewith very long time-scales. (2) Biological processes that will result if life adapts itself tolow ambient temperatures according to a postulated scaling law. (3) Communication byradio between life forms existing in dierent parts of the universe. The general conclusionof the analysis is that an open universe need not evolve into a state of permanentquiescence. Life and communication can continue for ever, utilizing a nite store of energy,if the assumed scaling laws are valid.

(*) This material was originally presented as four lectures, the "James Arthur Lectureson Time and its Mysteries" at New York University, Autumn 1978. The rst lectureis addressed to a general audience, the other three to an audience of physicists andastronomers.

CONTENTS

Lecture I. Philosophy

Lecture II. Physics

A. Stellar evolution

B. Detachment of planets from stars

C. Detachment of stars from galaxies

D. Decay of orbits by gravitational radiation

E. Decay of black holes by the Hawking process

F. Matter is liquid at zero temperature

G. All matter decays to iron

H. Collapse of iron star to neutron star

I. Collapse of ordinary matter to black hole

Lecture III. Biology

Lecture IV. Communication

References

LECTURE I. PHILOSOPHY

A year ago Steven Weinberg published an excellent book, The First Three Minutes,(Weinberg, 1977), explaining to a lay audience the state of our knowledge about thebeginning of the universe. In his sixth chapter he describes in detail how progress inunderstanding and observing the universe was delayed by the timidity of theorists.

"This is often the way it is in physics our mistake is not that we take our theoriestoo seriously, but that we do not take them seriously enough. It is always hard to realizethat these numbers and equations we play with at our desks have something to do withthe real world. Even worse, there often seems to be a general agreement that certainphenomena are just not t subjects for respectable theoretical and experimental eort.Alpher, Herman and Gamow (1948) deserve tremendous credit above all for being willing

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to take the early universe seriously, for working out what known physical laws have tosay about the rst three minutes. Yet even they did not take the nal step, to convincethe radio astronomers that they ought to look for a microwave radiation background.The most important thing accomplished by the ultimate discovery of the 3 K radiationbackground (Penzias and Wilson, 1965) was to force all of us to take seriously the ideathat there was an early universe."

Thanks to Penzias and Wilson, Weinberg and others, the study of the beginning ofthe universe is now respectable. Professional physicists who investigate the rst threeminutes or the rst microsecond no longer need to feel shy when they talk about theirwork. But the end of the universe is another matter. I have searched the literature forpapers about the end of the universe (Rees, 1969; Davies, 1973; Islam, 1977 and 1979;Barrow and Tipler, 1978). This list is certainly not complete. But the striking thingabout these papers is that they are written in an apologetic or jocular style, as if theauthors were begging us not to take them seriously. The study of the remote future stillseems to be as disreputable today as the study of the remote past was thirty years ago.I am particularly indebted to Jamal Islam for an early draft of his 1977 paper whichstarted me thinking seriously about the remote future. I hope with these lectures tohasten the arrival of the day when eschatology, the study of the end of the universe, willbe a respectable scientic discipline and not merely a branch of theology.

Weinberg himself is not immune to the prejudices that I am trying to dispel. At theend of his book about the past history of the universe, he adds a short chapter aboutthe future. He takes 150 pages to describe the rst three minutes, and then dismisses thewhole of the future in ve pages. Without any discussion of technical details, he sums uphis view of the future in twelve words:

"The more the universe seems comprehensible, the more it also seems pointless."

Weinberg has here, perhaps unintentionally, identied a real problem. It is impossibleto calculate in detail the long-range future of the universe without including the eectsof life and intelligence. It is impossible to calculate the capabilities of life and intelligencewithout touching, at least peripherally, philosophical questions. If we are to examinehow intelligent life may be able to guide the physical development of the universe forits own purposes, we cannot altogether avoid considering what the values and purposesof intelligent life may be. But as soon as we mention the words value and purpose, werun into one of the most rmly entrenched taboos of twentieth-century science. Hear thevoice of Jacques Monod (1970), high priest of scientic rationality, in his book Chance

and Necessity :

"Any mingling of knowledge with values is unlawful, forbidden."

Monod was one of the seminal minds in the owering of molecular biology in thiscentury. It takes some courage to defy his anathema. But I will defy him, and encourageothers to do so. The taboo against mixing knowledge with values arose during thenineteenth century out of the great battle between the evolutionary biologists led byThomas Huxley and the churchmen led by Bishop Wilberforce. Huxley won the battle,but a hundred years later Monod and Weinberg were still ghting Bishop Wilberforce'sghost. Physicists today have no reason to be afraid of Wilberforce's ghost. If our analysisof the long-range future leads us to raise questions related to the ultimate meaning andpurpose of life, then let us examine these questions boldly and without embarrassment.If our answers to these questions are naive and preliminary, so much the better for thecontinued vitality of our science.

I propose in these lectures to explore the future as Weinberg in his book exploredthe past. My arguments will be rough and simple but always quantitative. The aim is toestablish numerical bounds within which the destiny of the universe must lie. I shall makeno further apology for mixing philosophical speculations with mathematical equations.

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The two simplest cosmological models (Weinberg, 1972) describe a uniform zero-pressure universe which may be either closed or open. The closed universe has its geometrydescribed by the metric

ds2 = R2[dψ2 − dχ2 − sin2 χdΩ2], (1)

where χ is a space coordinate moving with the matter, ψ is a time coordinate related tophysical time t by

t = T0 (ψ − sinψ), (2)

and R is the radius of the universe given by

R = c T0 (1− cosψ). (3)

The whole universe is represented in terms of the coordinates (ψ, χ) by a nite rectangularbox

0 < ψ < 2π, 0 < χ < π. (4)

This universe is closed both in space and in time. Its total duration is

2πT0, (5)

where T0 is a quantity that is in principle measurable. If our universe is described by thismodel, then T0 must be at least 1010 years.

The simple model of a uniform zero-pressure open universe has instead of (1) themetric

ds2 = R2[dψ2 − dχ2 − sinh2 χdΩ2], (6)

where nowt = T0 (sinhψ − ψ), (7)

R = c T0 (coshψ − 1), (8)

and the coordinates (ψ, χ) extend over an innite range

0 < ψ <∞, 0 < χ <∞. (9)

The open universe is innite both in space and in time.The models (1) and (6) are only the simplest possibilities. Many more complicated

models can be found in the literature. For my purpose it is sucient to discuss (1) and (6)as representative of closed and open universes. The great question, whether our universeis in fact closed or open, will before long be settled by observation. I do not say moreabout this question, except to remark that my philosophical bias strongly favors an openuniverse and that the observational evidence does not exclude it (Gott, Gunn, Schramm,and Tinsley, 1974 and 1976).

The prevailing view (Weinberg, 1977) holds the future of open and closed universesto be equally dismal. According to this view, we have only the choice of being fried in aclosed universe or frozen in an open one. The end of the closed universe has been studiedin detail by Rees (1969). Regrettably I have to concur with Rees' verdict that in thiscase we have no escape from frying. No matter how deep we burrow into the earth toshield ourselves from the ever-increasing fury of the blue-shifted background radiation,we can only postpone by a few million years our miserable end. I shall not discuss theclosed universe in detail, since it gives me a feeling of claustrophobia to imagine our wholeexistence conned within the box (4). I only raise one question which may oer us a thinchance of survival. Supposing that we discover the universe to be naturally closed anddoomed to collapse, is it conceivable that by intelligent intervention, converting matter

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into radiation and causing energy to ow purposefully on a cosmic scale, we could breakopen a closed universe and change the topology of space-time so that only a part of itwould collapse and another part of it would expand forever? I do not know the answerto this question. If it turns out that the universe is closed, we shall still have about 1010

years to explore the possibility of a technological x that would burst it open.I am mainly interested in the open cosmology, since it seems to give enormously

greater scope for the activities of life and intelligence. Horizons in the open cosmologyexpand indenitely. To be precise, the distance to the horizon in the metric (6) is

d = Rψ, (10)

with R given by (8), and the number of galaxies visible within the horizon is

N = N0(sinh 2ψ − 2ψ), (11)

where N0 is a number of the order of 1010.Comparing (11) with (7), we see that the number of visible galaxies varies with t2 at

late times. It happens by a curious numerical accident that the angular size of a typicalgalaxy at time t is

δ ∼ 105 t−1 rad, (12)

with t measured in years. Since (11) and (7) give

N ∼ 10−10t2, Nδ2 ∼ 1, (13)

it turns out that the sky is always just lled with galaxies, no matter how far into thefuture we go. As the apparent size of each galaxy dwindles, new galaxies constantlyappear at the horizon to ll in the gaps. The light from the distant galaxies will bestrongly red-shifted. But the sky will never become empty and dark, if we can tune oureyes to longer and longer wavelengths as time goes on.

I shall discuss three principal questions within the framework of the open universewith the metric (6).

(1) Does the universe freeze into a state of permanent physical quiescence as itexpands and cools?

(2) Is it possible for life and intelligence to survive indenitely?(3) Is it possible to maintain communication and transmit information across the

constantly expanding distances between galaxies?These three questions will be discussed in detail in Lectures 2, 3 and 4. Tentatively,

I shall answer them with a no, a yes, and a maybe. My answers are perhaps only areection of my optimistic philosophical bias. I do not expect everybody to agree withthe answers. My purpose is to start people thinking seriously about the questions.

If, as I hope, my answers turn out to be right, what does it mean? It means thatwe have discovered in physics and astronomy an analog to the theorem of Godel (1931)in pure mathematics. Godel proved [see Nagel and Newman (1956)] that the world ofpure mathematics is inexhaustible; no nite set of axioms and rules of inference canever encompass the whole of mathematics; given any nite set of axioms, we can ndmeaningful mathematical questions which the axioms leave unanswered. I hope that ananalogous situation exists in the physical world. If my view of the future is correct, itmeans that the world of physics and astronomy is also inexhaustible; no matter howfar we go into the future, there will always be new things happening, new informationcoming in, new worlds to explore, a constantly expanding domain of life, consciousness,and memory.

When I talk in this style, I am mixing knowledge with values, disobeying Monod'sprohibition. But I am in good company. Before the days of Darwin and Huxley and Bishop

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Wilberforce, in the eighteenth century, scientists were not subject to any taboo againstmixing science and values. When Thomas Wright (1750), the discoverer of galaxies,announced his discovery, he was not afraid to use a theological argument to supportan astronomical theory.

"Since as the Creation is, so is the Creator also magnied, we may conclude inconsequence of an innity, and an innite all-active power, that as the visible creation issupposed to be full of siderial systems and planetary worlds, so on, in like similar manner,the endless immensity is an unlimited plenum of creations not unlike the known.... Thatthis in all probability may be the real case, is in some degree made evident by themany cloudy spots, just perceivable by us, as far without our starry Regions, in whichtho' visibly luminous spaces, no one star or particular constituent body can possiblybe distinguished; those in all likelyhood may be external creation, bordering upon theknown one, too remote for even our telescopes to reach."

Thirty-ve years later, Wright's speculations were conrmed by William Herschel'sprecise observations. Wright also computed the number of habitable worlds in our galaxy:

"In all together then we may safely reckon 170,000,000, and yet be much withincompass, exclusive of the comets which I judge to be by far the most numerous part ofcreation."

His statement about the comets may also be correct, although he does not tell ushow he estimated their number. For him the existence of so many habitable worlds wasnot just a scientic hypothesis but a cause for moral reection:

"In this great celestial creation, the catastrophy of a world, such as ours, or even thetotal dissolution of a system of worlds, may be possibly be no more to the great Authorof Nature, than the most common accident in life with us, and in all probability suchnal and general Doomsdays may be as frequent there, as even Birthdays or mortalitywith us upon the earth. This idea has something so cheerful in it, that I know I cannever look upon the stars without wondering why the whole world does not becomeastronomers; and that endowed with sense and reason should neglect a science they arenaturally so much interested in, and so capable of enlarging their understanding, as nextto a demonstration must convince them of their immortality, and reconcile them to allthose little diculties incident to human nature, without the least anxiety.

"All this the vast apparent provision in the starry mansions seem to promise: Whatought we then not to do, to preserve our natural birthright to it and to merit suchinheritance, which alas we think created all to gratify alone a race of vain-glorious giganticbeings, while they are conned to this world, chained like so many atoms to a grain ofsand."

There speaks the eighteenth century. But Steven Weinberg says, "The more theuniverse seems comprehensible, the more it also seems pointless." If Weinberg is speakingfor the twentieth century, I prefer the eighteenth.

LECTURE II. PHYSICS

In this lecture, following Islam (1977), I investigate the physical processes that willoccur in an open universe over very long periods of time. I consider the natural universeundisturbed by eects of life and intelligence. Life and intelligence will be discussed inlectures 3 and 4.

Two assumptions underlie the discussion. (1) The laws of physics do not change withtime. (2) The relevant laws of physics are already known to us. These two assumptionswere also made by Weinberg (1977) in his description of the past. My justication formaking them is the same as his. Whether or not we believe that the presently knownlaws of physics are the nal and unchanging truth, it is illuminating to explore the

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consequences of these laws as far as we can reach into the past or the future. It is betterto be too bold than too timid in extrapolating our knowledge from the known into theunknown. It may happen again, as it happened with the cosmological speculations ofAlpher, Herman, and Gamow (1948), that a naive extrapolation of known laws into newterritory will lead us to ask important new questions.

I have summarized elsewhere (Dyson, 1972, 1978) the evidence supporting the hypothesisthat the laws of physics do not change. The most striking piece of evidence was discoveredrecently by Shlyakhter (1976) in the measurements of isotope ratios in ore samples takenfrom the natural ssion reactor that operated about 2 billion years ago in the Oklouraniummine in Gabon (Maurette, 1976). The crucial quantity is the ratio (149Sm/147Sm)between the abundances of two light isotopes of samarium which are not ssion products.In normal samarium this ratio is about 0.9; in the Oklo reactor it is about 0.02. Evidentlythe 149Sm has been heavily depleted by the dose of thermal neutrons to which it wasexposed during the operation of the reactor. If we measure in a modern reactor thethermal neutron capture cross section of 149Sm, we nd the value 55 kb, dominated by astrong capture resonance at a neutron energy of 0.1 eV. A detailed analysis of the Okloisotope ratio leads to the conclusion that the 149Sm cross section was in the range 55±8 kbtwo billion years ago. This means that the position of the capture resonance cannot haveshifted by as much as 0.02 eV over 2 · 109 yr. But the position of this resonance measuresthe dierence between the binding energies of the 149Sm ground state and of the 150Smcompound state into which the neutron is captured. These binding energies are each ofthe order of 109 eV and depend in a complicated way upon the strengths of nuclear andCoulomb interactions. The fact that the two binding energies remained in balance to anaccuracy of two parts in 1011 over 2 · 109 yr indicates that the strength of nuclear andCoulomb forces cannot have varied by more than a few parts in 1018 per year. This is byfar the most sensitive test that we have yet found of the constancy of the laws of physics.The fact that no evidence of change was found does not, of course, prove that the lawsare strictly constant. In particular, it does not exclude the possibility of a variation instrength of gravitational forces with a time scale much shorter than 1018 yr. For the sakeof simplicity, I assume that the laws are strictly constant. Any other assumption wouldbe more complicated and would introduce additional arbitrary hypotheses.

It is in principle impossible for me to bring experimental evidence to support thehypothesis that the laws of physics relevant to the remote future are already knownto us. The most serious uncertainty aecting the ultimate fate of the universe is thequestion whether the proton is absolutely stable against decay into lighter particles. Ifthe proton is unstable, all matter is transitory and must dissolve into radiation. Someserious theoretical arguments have been put forward (Zeldovich, 1977; Barrow and Tipler,1978; Feinberg, Goldhaber, and Steigman, 1978) supporting the view that the protonshould decay with a long half-life, perhaps through virtual processes involving blackholes. The experimental limits on the rate of proton decay (Kropp and Reines, 1965) donot exclude the existence of such processes. Again on grounds of simplicity, I disregardthese possibilities and suppose the proton to be absolutely stable. I will discuss in detaillater the eect of real processes involving black holes on the stability of matter in bulk.

I am now ready to begin the discussion of physical processes that will occur in the opencosmology (6), going successively to longer and longer timescales. Classical astronomicalprocesses come rst, quantum-mechanical processes later.

Note added in proof. Since these lectures were given, a spate of papers has appeareddiscussing grand unication models of particle physics in which the proton is unstable(Nanopoulos, 1978; Pati, 1979; Turner and Schramm, 1979).

A. Stellar evolution

The longest-lived low-mass stars will exhaust their hydrogen fuel, contract into white

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dwarf congurations, and cool down to very low temperatures, within times of the orderof 1014 years. Stars of larger mass will take a shorter time to reach a cold nal state,which may be a white dwarf, a neutron star, or a black hole conguration, depending onthe details of their evolution.

B. Detachment of planets from starsThe average time required to detach a planet from a star by a close encounter with

a second star isT = (ρ V σ)−1, (14)

where ρ is the density of stars in space, V the mean relative velocity of two stars, andsigma the cross section for an encounter resulting in detachment. For the earth-sunsystem, moving in the outer regions of the disk of a spiral galaxy, approximate numericalvalues are

ρ = 3 · 10−41 km−3, (15)V = 50 k/sec, (16)σ = 2 · 1016 km2, (17)T = 1015 yr.(18)

The time scale for an encounter causing serious disruption of planetary orbits will beconsiderably shorter than 1015 yr.

C. Detachment of stars from galaxiesThe dynamical evolution of galaxies is a complicated process, not yet completely

understood. I give here only a very rough estimate of the time scale. If a galaxy of Nstars of massM in a volume of radius R, their root-mean-square velocity will be of order

V =[GNM

R

]1/2. (19)

The cross section for a close encounter between two stars, changing their directions ofmotion by a large angle, is

σ =(GM

V 2

)2

=(R

N

)2

. (20)

The average time that a star spends between two close encounters is

T = (ρ V σ)−1 =

(NR3

GM

)1/2

. (21)

If we are considering a typical large galaxy with N = 1011, R = 3 · 1017 km, then

T = 1019 yr. (22)

Dynamical relaxation of the galaxy proceeds mainly through distant stellar encounterswith a time scale

TN = T (logN)−1 = 1018 yr. (23)

The combined eect of dynamical relaxation and close encounters is to produce a collapseof the central regions of the galaxy into a black hole, together with an evaporation ofstars from the outer regions. The evaporated stars achieve escape velocity and becomedetached from the galaxy after a time of the order of 1019 yr. We do not know whatfraction of the mass of the galaxy ultimately collapses and what fraction escapes. Thefraction escaping probably lies between 90% and 99%.

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The violent events which we now observe occurring in the central regions of manygalaxies are probably caused by a similar process of dynamical evolution oprating on amuch shorter time scale. According to (21), the time scale for evolution and collapse willbe short if the dynamical units are few and massive, for example compact star clustersand gas clouds rather than individual stars. The long time scale (22) applies to a galaxycontaining no dynamical units larger than individual stars.

D. Decay of orbits by gravitational radiationIf a mass is orbiting around a xed center with velocity V , period P , and kinetic

energy E, it will have energy by gravitational radiation at a rate of order

Eg = (V/c)5 (E/P ). (24)

Any gravitationally bound system of objects orbiting around each other will decay bythis mechanism of radiation drag with a time scale

tg = (c/V )5 P. (25)

For the earth orbiting around the sun, the gravitational radiation time scale is

Tg = 1020 yr. (26)

Since this is much longer than (18), the earth will almost certainly escape from the sunbefore gravitational radiation can pull it inward. But if it should happen that the sunshould escape from the galaxy with the earth still attached to it, then the earth willultimately coalesce with the sun after a time of order (26).

The orbits of the stars in a galaxy will also be decaying by gravitational radiationwith time scale (25), where P is now the period of their galactic orbits. For a galaxy likeour own, with V = 200 km/sec and P = 2 · 108 yr, the time scale is

Tg = 1024 yr. (27)

This is again much longer than (22), showing that dynamical relaxation dominatesgravitational radiation in the evolution of galaxies.

E. Decay of black holes by the Hawking processAccording to Hawking (1975), every black hole of mass M decays by emission of

thermal radiation and nally disappears after a time

T = (G2M3/~ c4). (28)

For a black hole of one solar mass the lifetime is

T = 1064 yr. (29)

Black holes of galactic mass will have lifetimes extending up to 10100 yr. At the end ofits life, every black hole will emit about 1031 erg of high-temperature radiation. The coldexpanding universe will be illuminated by occasional reworks for a very long time.

F. Matter is liquid at zero temperatureI next discuss a group of physical processes which occur in ordinary matter at zero

temperature as a result of quantum-mechanical barrier penetration. The lifetimes forsuch processes are given by the Gamow formula

T = exp(S)T0, (30)

where T0 is a natural vibration period of the system, and S is the action integral

S = (2/~)∫

(2MU(x))1/2 dx. (31)

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Here x is a coordinate measuring the state of the system as it goes across the barrier,and U(x) is the height of the barrier as a function of x. To obtain a rough estimate ofS, I replace (31) by

S = (8MUd2/~2)1/2, (32)

where d is the thickness, and U the average height of the barrier, and M is the mass ofthe object that is moving across it. I shall consider processes for which S is large, so thatthe lifetime (30) is extremely long.

As an example, consider the behavior of a lump of matter, a rock or a planet, after ithas cooled to zero temperature. Its atoms are frozen into an apparently xed arrangementby the forces of cohesion and chemical bonding. But from time to time the atoms will moveand rearrange themselves, crossing energy barriers by quantum-mechanical tunneling.The height of the barrier will typically be of the order of a tenth of a Rydberg unit,

U = (1/20)(e4m/~2), (33)

and the thickness will be of the order of a Bohr radius

d = (~2/me2), (34)

where m is the electron mass. The action integral (32) is then

S = (2Amp/5m)1/2 = 27A1/2, (35)

where mp is the proton mass, and A is the atomic weight of the moving atom. For aniron atom with A = 56, S = 200, and (30) gives

T = 1065 yr. (36)

Even the most rigid materials cannot preserve their shapes or their chemical structuresfor times long compared with (36). On a time scale of 1065 yr, every piece of rock behaveslike a liquid, owing into a spherical shape under the inuence of gravity. Its atoms andmolecules will be ceaselessly diusing around like the molecules in a drop of water.

G. All matter decays to ironIn matter at zero temperature, nuclear as well as chemical reactions will continue to

occur. Elements heavier than iron will decay to iron b various processes such as ssionand alpha emission. Elements lighter than iron will combine by nuclear fusion reactions,building gradually up to iron. Consider for example the fusion reaction in which twonuclei of atomic weight 1

2 A, charge12 Z combine to form a nucleus (A,Z). The Coulomb

repulsion of the two nuclei is eectively screened by electrons until they come within adistance

d = Z−1/2(~2/me2) (37)

of each other. The Coulomb barrier has thickness d and height

U = (Z2 e2/4d) = 12 Z

7/3(e4m/~2). (38)

The reduced mass for the relative motion of the two nuclei is

M =14AMp. (39)

The action integral (32) then becomes

S =(

12 AZ

5/3(mp/m))1/2

= 30A1/2Z5/6. (40)

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For two nuclei combining to form iron, Z = 26, A = 56, S = 3500, and

T = 101500 yr. (41)

On the time scale (41), ordinary matter is radioactive and is constantly generating nuclearenergy.

H. Collapse of iron star to neutron starAfter the time (41) has elapsed, most of the matter in the universe is in the form

of ordinary low-mass stars that have settled down into white dwarf congurations andbecome cold spheres of pure iron. But an iron star is still not in its state of lowestenergy. It could release a huge amount of energy if it could collapse into a neutron starconguration. To collapse, it has only to penetrate a barrier of nite height and thickness.It is an interesting question, whether there is an unsymmetrical mode of collapse passingover a lower saddle point than the symmetric mode. I have not been able to nd aplausible unsymmetric mode, and so I assume the collapse to be spherically symmetrical.In the action integral (31), the coordinate x will be the radius of the star, and the integralwill extend from r, the radius of a neutron star, to R, the radius of the iron star fromwhich the collapse begins. The barrier height U(x) will depend on the equation of state ofthe matter, which is very uncertain when x is close to r. Fortunately the equation of stateis well known over the major part of the range of integration, when x is large compared tor and the main contribution to U(x) is the energy of nonrelativistic degenerate electrons

U(x) = (N5/3~2/2mx2), (42)

where N is the number of electrons in the star.The integration over x in (31) gives algorithm

log(R/R0), (43)

where R0 is the radius at which the electrons become relativistic and the formula (42)fails. For low-mass stars the logarithm will be of the order of unity, and the part of theintegral coming from the relativistic region x < R0 will also be of the order of unity. Themass of the star is

M = 2N mp. (44)

I replace the logarithm (43) by unity and obtain for the action integral (31) the estimate

S = N4/3 (8mp/m)1/2 = 120N4/3. (45)

The lifetime is then by (30)T = exp(120N4/3)T0. (46)

For a typical low-mass star we have

N = 1056, S = 1077, T = 101076yr. (47)

In (46) it is completely immaterial whether T0 is a small fraction of a second or a largenumber of years.

We do not know whether every collapse of an iron star into a neutron star will producea supernova explosion. At the very least, it will produce a huge outburst of energy in theform of neutrinos and a modest burst of energy in the form of x rays and visible light.The universe will still be producing occasional reworks after times as long as (47).

I. Collapse of ordinary matter to black holesThe long lifetime (47) of iron stars is only correet if they do not collapse with a shorter

lifetime into black holes. For collapse of any piece of bulk matter into a black hole, the

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same formulae apply as for collapse into a neutron star. The only dierence is that theintegration in the action integral (31) now extends down to the black hole radius insteadof to the neutron star radius. The main part of the integral comes from larger values ofx and is the same in both cases. The lifetime for collapse into a black hole is thereforestill given by (46). But there is an important change in the meaning of N. If small blackholes are possible, a small part of a star can collapse by itself into a black hole. Once asmall black hole has been formed, it will in a short time swallow the rest of the star. Thelifetime for collapse of any star is then given by

T = exp(120N4/3B )T0, (48)

where NB is the number of electrons in a piece of iron of mass equal to the minimummass MB of a black hole. The lifetime (48) is the same for any piece of matter of massgreater than MB. Matter in pieces with mass smaller than MB is absolutely stable. Fora more complete discussion of the problem of collapse into black holes, see Harrison,Thorne, Wakano, and Wheeler (1965).

The numerical value of the lifetime (48) depends on the value of MB. All that weknow for sure is

0 6 MB 6 Mc, (49)

where

Mc = (~c/G)3/2m−2p = 4.1033 g (50)

is the Chandrasekhar mass. Black holes must exist for every mass larger thanMc, becausestars with mass larger than Mc have no stable nal state and must inevitably collapse.

Four hypotheses concerning MB have been put forward:(i) MB = 0. Then black holes of arbitrarily small mass exist and the formula (48) is

meaningless. In this case all matter is unstable with a comparatively short lifetime, assuggested by Zeldovich (1977).

(ii) MB is equal to the Planck mass

MB = MPL = (~ c/G)1/2 = 2 · 10−5 g. (51)

This value ofMB is suggested by Hawking's theory of radiation from black holes (Hawking,1975), according to which every black hole loses mass until it reaches a mass of orderMPL, at which point it disappears in a burst of radiation. In this case (48) gives

NB = 1019, T = 101026yr. (52)

(iii) MB is equal to the quantum mass

MB = MQ = (~ c/GmP ) = 3 · 1014 g, (53)

as suggested by Harrison, Thorne, Wakano, and Wheeler (1965). Here MQ is the mass ofthe smallest black hole for which a classical theory is meaningful. Only for masses largerthan MQ can we consider the barrier penetration formula (31) to be physically justied.If (53) holds, then

NB = 1038, T = 101032yr. (54)

(iv)MB is equal to the Chandrasekhar mass (50). In this case the lifetime for collapseinto a black hole is of the same order as the lifetime (47) for collapse into a neutron star.

The long-range future of the universe depends crucially on which of these four alternativesis correct. If (iv) is correct, stars may collapse into black holes and dissolve into pureradiation, but masses of planetary size exist forever. If (iii) is correct, planets will

12

disappear with the lifetime (54), but material objects with masses up to a few milliontons are stable. If (ii) is correct, human-sized objects will disappear with the lifetime(52), but dust grains with diameter less than about 100 mu will last for ever. If (i) iscorrect, all material objects disappear and only radiation is left.

If I were compelled to choose one of the four alternatives as more likely than theothers, I would choose (ii). I consider (iii) and (iv) unlikely because they are inconsistentwith Hawking's theory of black-hole radiation. I nd (i) implausible because it is dicultto see why a proton should not decay rapidly if it can decay at all. But in our presentstate of ignorance, none of the four possibilities can be excluded.

The results of this lecture are summarized in Table I. This list of time scales ofphysical processes makes no claim to be complete. Undoubtedly many other physicalprocesses will be occurring with time scales as long as, or longer than, those I have listed.The main conclusion I wish to draw from my analysis is the following: So far as we canimagine into the future, things continue to happen. In the open cosmology, history hasno end.

TABLE I. Summary of time scales.

Closed Universe

Total duration 1011 yr

Open Universe

Low-mass stars cool o 1014 yr

Planets detached from stars 1015 yr

Stars detached from galaxies 1019 yr

Decay of orbits by gravitational radiation 1020 yr

Decay of black holes by Hawking process 1064 yr

Matter liquid at zero temperature 1065 yr

All matter decays to iron 101500 yrCollapse of ordinary matter to

black hole [alternative (ii)] 101026

yrCollapse of stars to neutron stars

or black holes [alternative (iv)] 101076

yr

LECTURE III. BIOLOGY

Looking at the past history of life, we see that it takes about 106 years to evolve anew species, 107 years to evolve a genus, 108 years to evolve a class, 109 years to evolve aphylum, and less than 1010 years to evolve all the way from the primeval slime to HomoSapiens. If life continues in this fashion in the future, it is impossible to set any limit tothe variety of physical forms that life may assume. What changes could occur in the next1010 years to rival the changes of the past? It is conceivable that in another 1010 yearslife could evolve away from esh and blood and become embodied in an interstellar blackcloud (Hoyle, 1957) or in a sentient computer (Capek, 1923).

Here is a list of deep questions concerning the nature of life and consciousness.

(i) Is the basis of consciousness matter or structure?

(ii) Are sentient black clouds, or sentient computers, possible?

(iii) Can we apply scaling laws in biology?

These are questions that we do not know how to answer. But they are not in principleunanswerable. It is possible that they will be answered fairly soon as a result of progressin experimental biology.

13

Let me spell out more explicitly the meaning of question (i). My consciousness issomehow associated with a collection of organic molecules inside my head. The questionis, whether the existence of my consciousness depends on the actual substance of aparticular set of molecules or whether it only depends on the structure of the molecules.In other words, if I could make a copy of my brain with the same structure but usingdierent materials, would the copy think it was me?

If the answer to question (i) is "matter", then life and consciousness can never evolveaway from esh and blood. In this case the answers to questions (ii) and (iii) are negative.Life can then continue to exist only so long as warm environments exist, with liquid waterand a continuing supply of free energy to support a constant rate of metabolism. In thiscase, since a galaxy has only a nite supply of free energy, the duration of life is nite.As the universe expands and cools, the sources of free energy that life requires for itsmetabolism will ultimately be exhausted.

Since I am a philosophical optimist, I assume as a working hypothesis that theanswer to question (i) is "structure". Then life is free to evolve into whatever materialembodiment best suits its purposes. The answers to questions (ii) and (iii) are armative,and a quantitative discussion of the future of life in the universe becomes possible. Ifit should happen, for example, that matter is ultimately stable against collapse intoblack holes only when it is subdivided into dust grains a few microns in diameter, thenthe preferred embodiment for life in the remote future must be something like Hoyle'sblack cloud, a large assemblage of dust grains carrying positive and negative charges,organizing itself and communicating with itself by means of electromagnetic forces. Wecannot imagine in detail how such a cloud could maintain the state of dynamic equilibriumthat we call life. But we also could not have imagined the architecture of a living cell ofprotoplasm if we had never seen one.

For a quantitative description of the way life may adapt itself to a cold environment, Ineed to assume a scaling law that is independent of any particular material embodimentthat life may nd for itself. The following is a formal statement of my scaling law:

Biological Scaling Hypothesis. If we copy a living creature, quantum state by quantum

state, so that the Hamiltonian of the copy is

Hc = λU H U−1, (55)

where H is the Hamiltonian of the creature, U is a unitary operator, and λ is a positive

scaling factor, and if the environment is similarly copied so that the temperatures of the

environments of the creature and the copy are respectively T and λT , then the copy is

alive, subjectively identical to the original creature, with all its vital functions reduced in

speed by the same factor λ.

The structure of the Schrodinger equation, with time and energy appearing as conjugatevariables, makes the form of this scaling hypothesis plausible. It is at present a purelytheoretical hypothesis, not susceptible to any experimental test. To avoid misunderstanding,I should emphasize that the scaling law does not apply to the change of the metabolic rateof a given organism as a function of temperature. For example, when a snake or a lizardchanges its temperature, its metabolic rate varies exponentially rather than linearly withT . The linear scaling law applies to an ensemble of copies of a snake, each copy adaptedto a dierent temperature. It does not apply to a particular snake with varying T .

From this point on, I assume the scaling hypothesis to be valid and examine itsconsequences for the potentialities of life. The rst consequence is that the appropriatemeasure of time as experienced subjectively by a living creature is not physical time t

14

but the quantity

u(t) = f

t∫0

θ(t′) dt′, (56)

where θ(t) is the temperature of the creature and f = (300 deg sec)−1 is a scale factorwhich it is convenient to introduce so as to make u dimensionless. I call u "subjectivetime". The second consequence of the scaling law is that any creature is characterized bya quantity Q which measures its rate of entropy production per unit of subjective time.If entropy is measured in information units or bits, and if u is measured in "momentsof consciousness", then Q is a pure number expressing the amount of information thatmust be processed in order to keep the creature alive long enough to say "Cogito, ergosum". I call Q the "complexity"of the creature. For example, a human being dissipatesabout 200 W of power at a temperature of 300 K, with each moment of consciousnesslasting about a second. A human being therefore has

Q = 1023 bits. (57)

This Q is a measure of the complexity of the molecular structures involved in a singleact of human awareness. For the human species as a whole,

Q = 1033 bits. (58)

a number which tells us the order of magnitude of the material resources required formaintenance of an intelligent society.

A creature or a society with given Q and given temperature theta will dissipate energyat a rate

m = k f Qθ2. (59)

Here m is the metabolic rate measured in ergs per second, k is Boltzmann's constant,and f is the coecient appearing in (56). It is important that m varies with the squareof theta, one factor theta coming from the relationship between energy and entropy, theother factor θ coming from the assumed temperature dependence of the rate of vitalprocesses.

I am assuming that life is free to choose its temperature θ(t) so as to maximize itschances of survival. There are two physical constraints on θ(t). The rst constraint is thatθ(t) must always be greater than the temperature of the universal background radiation,which is the lowest temperature available for a heat sink. That is to say

θ(t) > aR−1, a = 3 · 1028 deg cm, (60)

where R is the radius of the universe, varying with t according to (7) and (8). At thepresent time the condition (60) is satised with a factor of 100 to spare. The secondconstraint on θ(t) is that a physical mechanism must exist for radiating away intospace the waste heat generated by metabolism. To formulate the second constraintquantitatively, I assume that the ultimate disposal of waste heat is by radiation andthat the only relevant form of radiation is electromagnetic.

There is an absolute upper limit

I(θ) < 2γ(Ne2/m ~2 c3)(k θ)3 (61)

on the power that can be radiated by a material radiator containing N electrons attemperature θ. Here

γ = max[x3 (ex − 1)−1] = 1.42 (62)

15

is the height of the maximum of the Planck radiation spectrum. Since I could not nd(61) in the textbooks, I give a quick proof, following the Handbuch article of Bethe andSaltpeter (1957). The formula for the power emitted by electric dipole radiation is

I(θ) =∑p

∫dΩ∑i

∑j

ρi(ω4ij/2π c

3)|Dij |2. (63)

Here p is the polarization vector of a photon emitted into the solid angle dΩ, i is theinitial and j the nal state of the radiator,

ρi = Z−1 exp(−Ei/kθ) (64)

is the probability that the radiator is initially in state i,

ωij = ~−1(Ei − Ej) (65)

is the frequency of the photon, and Dij is the matrix element of the radiator dipolemoment between states i and j. The sum (63) is taken only over pairs of states (i, j)with

Ei > Ej . (66)

Now there is an exact sum rule for dipole moments,∑i

ωij |Dij |2 = (1/2i) < DD − DD >jj= (N e2 ~/2m). (67)

But we have to be careful in using (67) to nd a bound for (63), since some of the termsin (67) are negative. The following trick works. In every term of (63), ωij is positive by(66), and so (62) gives

ρi ω3ij < γ ρi(kθ/~)3(exp(~ωij/kθ)− 1) = γ(ρj − ρi)(kθ/~)3. (68)

Therefore (63) implies

I(θ) < γ(kθ/~)3∑p

∫dΩ∑i

∑j

(ρj − ρi)(ωij2πc3)|Dij |

2. (69)

Now the summation indices (i, j) can be exchanged in the part of (69) involving ρi. Theresult is

I(θ) < γ(kθ/~)3∑p

∫dΩ∑i

∑j

ρj(ωij2πc3)|Dij |

2, (70)

with the summation now extending over all (i, j) whether (66) holds or not. The sumrule (67) can then be used in (70) and gives the result (61).

This proof of (61) assumes that all particles other than electrons have so large a massthat they are negligible in generating radiation. It also assumes that magnetic dipole andhigher multipole radiation is negligible. It is an interesting question whether (61) couldbe proved without using the dipole approximation (63).

It may at rst sight appear strange that the right side of (61) is proportional to θ3

rather than θ4, since the standard Stefan-Boltzmann formula for the power radiated bya black body is proportional to θ4. The Stefan-Boltzmann formula does not apply in thiscase because it requires the radiator to be optically thick. The maximum radiated powergiven by (61) can be attained only when the radiator is optically thin.

After this little digression into physics, I return to biology. The second constraint onthe temperature theta of an enduring form of life is that the rate of energy dissipation

16

(59) must not exceed the power (61) that can be radiated away into space. This constraintimplies a xed lower bound for the temperature,

kθ > (Q/N)ε = (Q/N)10−28 erg, (71)

ε = (137/2γ)(~f/k)mc2, (72)

θ > (Q/N)(ε/k) = (Q/N)10−12 deg. (73)

The ratio (Q/N) between the complexity of a society and the number of electrons at itsdisposal cannot be made arbitrarily small. For the present human species, with Q givenby (58) and

N = 1042 (74)

being the number of electrons in the earth's biosphere, the ratio is 10−9. As a societyimproves in mental capacity and sophistication, the ratio is likely to increase rather thandecrease. Therefore (73) and (59) imply a lower bound to the rate of energy dissipationof a society of a given complexity. Since the total store of energy available to a societyis nite, its lifetime is also nite. We have reached the sad conclusion that the slowingdown of metabolism described by my biological scaling hypothesis is insucient to allowa society to survive indenitely.

Fortunately, life has another strategy with which to escape from this impasse, namelyhibernation. Life may metabolize intermittently, but may continue to radiate waste heatinto space during its periods of hibernation. When life is in its active phase, it will be inthermal contact with its radiator at temperature θ. When life is hibernating, the radiatorwill still be at temperature θ but the life will be at a much lower temperature so thatmetabolism is eectively stopped.

Suppose then that a society spends a fraction g(t) of its time in the active phase anda fraction [1− g(t)] hibernating. The cycles of activity and hibernation should be shortenough so that g(t) and θ(t) do not vary appreciably during any one cycle. Then (56)and (59) no longer hold. Instead, subjective time is given by

u(t) = f

t∫0

g(t′) θ(t′) dt′, (74)

and the average rate of dissipation of energy is

m = k f Qg θ2. (75)

The constraint (71) is replaced by

θ(t) > (Q/N)(ε/k) g(t). (76)

Life keeps in step with the limit (61) on radiated power by lowering its duty cycle inproportion to its temperature.

As an example of a possible strategy for a long-lived society, we can satisfy theconstraints (60) and (76) by a wide margin if we take

g(t) = (θ(t)/θ0) = (t/t0)−α, (77)

where θ0 and t0 are the present temperature of life the present age of the universe. Theexponent alpha has to lie in the range

1/3 < α < 1/2, (78)

17

and for deniteness we take

α = 3/8. (79)

Subjective time then becomes by (74)

u(t) = A(t/t0)1/4, (80)

where

A = 4f θ0 t0 = 1018 (81)

is the present age of the universe measured in moments of consciousness. The averagerate of energy dissipation is by (75)

m(t) = k f Qθ20(t/t0)

−9/8. (82)

The total energy metabolized over all time from t0 to innity is

∞∫t0

m(t) dt = BQ, (83)

B = 2AK θ0 = 6 · 104 erg. (84)

This example shows that it is possible for life with the strategy of hibernation to achievesimultaneously its two main objectives. First, according to (80), subjective time is innite;although the biological clocks are slowing down and running intermittently as the universeexpands, subjective time goes on forever. Second, according to (83), the total energy

required for indenite survival is nite. The conditions (78) are sucient to make theintegral (83) convergent and the integral (74) divergent as t→∞.

According to (83) and (84), the supply of free energy required for the indenitesurvival of a society with the complexity (58) of the present human species, startingfrom the present time and continuing forever, is of the order

BQ = 6 · 1037 erg, (85)

about as much energy as the sun radiates in eight hours. The energy resources of a galaxywould be sucient to support indenitely a society with a complexity about 1024 timesgreater than our own.

These conclusions are valid in an open cosmology. It is interesting to examine thevery dierent situation that exists in a closed cosmology. If life tries to survive for aninnite subjective time in a closed cosmology, speeding up its metabolism as the universecontracts and the background radiation temperature rises, the relations (56) and (59)still hold, but physical time t has only a nite duration (5). If

τ = 2πT0 − t, (86)

the background radiation temperature

θR(t) = a(R(t))−1 (87)

is proportional to τ−2/3 as τ → 0, by virtue of (2) and (3). If the temperature θ(t) oflife remains close to θR as τ → 0, then the integral (56) is nite while the integral of(59) is innite. We have an innite energy requirement to achieve a nite subjectivelifetime. If θ(t) tends to innity more slowly than θR, the total duration of subjectivetime remains nite. If θ(t) tends to innity more rapidly than θR, the energy requirement

18

for metabolism remains innite. The biological clocks can never speed up fast enough tosqueeze an innite subjective time into a nite universe.

I return with a feeling of relief to the wide open spaces of the open universe. I donot need to emphasize the partial and preliminary character of the conclusions that Ihave presented in this lecture. I have only delineated in the crudest fashion a few of thephysical problems that life must encounter in its eort to survive in a cold universe. Ihave not addressed at all the multitude of questions that arise as soon as one tries toimagine in detail the architecture of a form of life adapted to extremely low temperatures.Do there exist functional equivalents in low-temperature systems for muscle, nerve, hand,voice, eye, ear, brain, and memory? I have no answers to these questions.

It is possible to say a little about memory without getting into detailed architecturalproblems, since memory is an abstract concept. The capacity of a memory can bedescribed quantitatively as a certain number of bits of information. I would like ourdescendants to be endowed not only with an innitely long subjective lifetime but alsowith a memory of endlessly growing capacity. To be immortal with a nite memory ishighly unsatisfactory; it seems hardly worthwhile to be immortal if one must ultimatelyerase all trace of one's origins in order to make room for new experience. There are twoforms of memory known to physicists, analog and digital. All our computer technologynowadays is based on digital memory. But digital memory is in principle limited incapacity by the number of atoms available for its construction. A society with nitematerial resources can never build a digital memory beyond a certain nite capacity.Therefore digital memory cannot be adequate to the needs of a life form planning tosurvive indenitely.

Fortunately, there is no limit in principle to the capacity of an analog memory builtout of a xed number of components in an expanding universe. For example, a physicalquantity such as the angle between two stars in the sky can be used as an analogmemory unit. The capacity of this memory unit is equal to the number of signicantbinary digits to which the angle can be measured. As the universe expands and thestars recede, the number of signicant digits in the angle will increase logarithmicallywith time. Measurements of atomic frequencies and energy levels can also in principlebe measured with a number of signicant gures proportional to (log t). Therefore animmortal civilization should ultimately nd ways to code its archives in an analog memorywith capacity growing like (log t). Such a memory will put severe constraints on the rateof acquisition of permanent new knowledge, but at least it does not forbid it altogether.

LECTURE IV. COMMUNICATION

In this last lecture I examine the problem of communication between two societiesseparated by a large distance in the open universe with metric (6). I assume that theycommunicate by means of electromagnetic signals. Without loss of generality I supposethat society A, moving along the world-line χ = 0, transmits, while society B, movingalong a world-line with the co-moving coordinate χ = η, receives. A signal transmittedby A when the time coordinate ψ = ξ will be received by B when ψ = ξ + η. If thetransmitted frequency is ω, the received frequency will be red-shifted to

ω′ =ω

1 + z=ωRARB

, (88)

RA = c T0 (cosh ξ − 1), (89)

RB = c T0 (cosh(ξ + η)− 1). (90)

The bandwidths B and B′ will be related by the same factor (1+z). The proper distancebetween A and B at the time the signal is received is dL = RB η. However, the area of

19

the sphere χ = η at the same instant is 4π d2T , with

dT = RB sinh η. (91)

If A transmits F photons per steradian in the direction of B, the number of photonsreceived by B will be

F ′ = (FΣ′/d2T ), (92)

where Σ′ is the eective cross section of the receiver.Now the cross section of a receiver for absorbing a photon of frequency ω′ is given by

a formula similar to (63) in the previous lecture

Σ′ =∑i

∑j

ρi(4π2ωji/~c)|Dij |

2δ(ωji − ω′), (93)

with Dij again a dipole matrix element between states i and j. When this is integrated

over all ω′, we obtain precisely the left side of the sum rule (67). The contribution fromnegative ω′ represents induced emission of a photon by the receiver. I assume that thereceiver is incoherent with the incident photon, so that induced emission is negligible.Then the sum rule gives

∞∫0

Σ′ dω′ = N ′(2π2 e2/mc), (94)

whereN ′ is the number of electrons in the receiver. If the receiver is tuned to the frequencyω′ with bandwidth B′, (94) gives

Σ′B′ 6 N ′ S0, (95)

S0 = (2π2e2/mc) = 0.167 cm2 sec−1. (96)

To avoid confusion of units, I measure both ω′ and B′ in radians per second rather thanin hertz. I assume that an advanced civilization will be able to design a receiver whichmakes (95) hold with equality. Then (92) becomes

F ′ = (F N ′ S0/d2T B

′). (97)

I assume that the transmitter contains N electrons which can be driven in phase so as toproduce a beam of radiation with angular spread of the order N−1/2. If the transmitteris considered to be an array of N dipoles with optimum phasing, the number of photonsper steradian in the beam is

F = (3N/8π)(E/~ω), (98)

where E is the total energy transmitted. The number of received photons is then

F ′ = (3N N ′E S0/8π~ω d2T B

′). (99)

We see at once from (99) that low frequencies and small bandwidths are desirablefor increasing the number of photons received. But we are interested in transmittinginformation rather than photons. To extract information eciently from a given numberof photons we should use a bandwidth equal to the detection rate,

B′ = (F ′/τB), B = (F ′/τA), (100)

where τB is the duration of the reception, and τA is the duration of the transmission.With this bandwidth, F ′ represents both the number of photons and also the number of

20

bits of information received. It is convenient to express τA and τB as a fraction of theradius of the universe at the times of transmission and reception

τA = (δRA/c), τB = (δRB/c). (101)

The conditionδ 6 1 (102)

then puts a lower bound on the bandwidth B. I shall also assume for simplicity that thefrequency ω is chosen to be as low as possible consistent with the bandwidth B, namely

ω = B, ω′ = B′. (103)

Then (99), (100), (101) give

F ′ =

[NN ′δ2E

(1 + z)(sinh2 η)Ec)

]1/3

, (104)

where by (96)Ec = (8π~c2/3S0) = (4/3π)137mc2 = 3 · 10−5 erg. (105)

We see from (104) that the quantity of information that can be transmitted from A to Bwith a given expenditure of energy does not decrease with time as the universe expandsand A and B move apart. The increase in distance is compensated by the decrease in theenergy cost of each photon and by the increase of receiver cross section with decreasingbandwidth.

The received signal is (104). We now have to compare it with the received noise. Thebackground noise in the universe at frequency ω can be described by an equivalent noisetemperature TN , so that the number of photons per unit bandwidth per steradian persquare centimeter per second is given by the RayleighJeans formula

I(ω) = (kTNω/4π3~c2). (106)

This formula is merely a denition of TN , which is in general a function of omega and t. Ido not assume that the noise has a Planck spectrum over the whole range of frequencies.Only a part of the noise is due to the primordial background radiation, which has a Planckspectrum with temperature θR. The primordial noise temperature θR varies inversely withthe radius of the universe,

(kθRR/~c) = Λ = 1029, (107)

with R given by (8). I assume that the total noise spectrum scales in the same way withradius as the universe expands, thus

(TN/θR) = f(x), x = (~ω/kθR), (108)

with f a universal function of x. When x is of the order of unity, the noise is dominatedby the primordial radiation, and f(x) takes the Planck form

f(x) = fp(x) = x(ex − 1)−1, x ∼ 1. (109)

But there will be strong deviations from (109) at large x (due to red-shifted starlight)and at small x (due to nonthermal radio sources). Without going into details, we can saythat f(x) is a generally decreasing function of x and tends to zero rapidly as x→∞.

The total energy density of radiation in the universe is

4π/c∫I(ω)~ω dω =

(kθR)4Iπ2~3c3

, (110)

21

with

I =∫ ∞

0f(x)x2 dx. (111)

The integral I must be convergent at both high and low frequencies. Therefore we cannd a numerical bound b such that

x3f(x) < b (112)

for all x. In fact (112) probably holds with b = 10 if we avoid certain discrete frequenciessuch as the 1420 MHz hydrogen line.

The number of noise photons received during the time τB by the receiver withbandwidth B′ and cross section Σ′ is

FN = 4πΣ′B′τBI(ω′). (113)

We substitute from (95), (96), (100), (103), (106), and (108) into (113) and obtain

FN = (2r0/γB)f N ′ F ′, (114)

wherer0 = (e2/mc2) = 3 · 10−13 cm, (115)

andγB = (hc/kθ′R) = Λ−1RB (116)

is the wavelength of the primordial background radiation at the time of reception. SinceF ′ is the signal, the signal-to-noise ratio is

RSN = (γB/2f N′ r0). (117)

In this formula, f is the noise-temperature ratio given by (108), N ′ is the number ofelectrons in the receiver, and r0, γB are given by (115), (116). Note that in calculating(117) we have not given the receiver any credit for angular discrimination, since the crosssection Σ′ given by (95) is independent of direction.

I now summarize the conclusions of the analysis so far. We have a transmitter and areceiver on the world-lines A and B, transmitting and receiving at times

tA = T0(sinh ξ − ξ), tB = T0(sinh(ξ + η)− (ξ + η)), (118)

According to (89) and (101),

τA = δ(dtA/dξ), τB = δ(dtB/dξ). (119)

It is convenient to think of the transmitter as permanently aimed at the receiver, andtransmitting intermittently with a certain duty cycle δ which may vary with ξ. Whenδ = 1 the transmitter is on all the time. The number F ′ of photons received in the timeτB can then be considered as a bit rate in terms of the variable ξ. In fact, F ′dξ is thenumber of bits received in the interval dξ. It is useful to work with the variable ξ sinceit maintains a constant dierence η between A and B.

From (100), (101), (103), (107), and (108) we derive a simple formula for the bit rate,

F ′ = Λx δ. (120)

The energy E transmitted in the time τA can also be considered as the rate of energytransmission per unit interval dξ. From (104) and (120) we nd

E = (Λ3/NN ′)(1 + z)(sinh2 η)x3 δEc. (121)

22

We are still free to choose the parameters x [determining the frequency ω by (108)] andδ, both of which may vary with ξ. The only constraints are (102) and the signal-to-noisecondition

RSN > 10, (122)

the signal-to-noise ratio being dened by (117). If I assume that (112) holds with b = 10,then (122) will be satised provided that

x > (G/r)1/3, (123)

with

G = (200r0/γp)N′(1 + z)−1 = 10−9N ′(1 + z)−1, (124)

r = (RA/Rp) = (cosh ξ − 1)/(cosh ξp − 1). (125)

Here γp, Rp and ξp are the present values of the background radiation wavelength, theradius of the universe, and the time coordinate ψ. It is noteworthy that the signal-to-noise condition (123) may be dicult to satisfy at early times when r is small, butgets progressively easier as time goes on and the universe becomes quieter. To avoid anextravagant expenditure of energy at earlier times, I choose the duty cycle δ to be smallat the beginning, increasing gradually until it reaches unity.

All the requirements are satised if we choose

x = max[(G/r)1/3, ξ−1/2], (126)

δ = min[(r/G)ξ−3/2, 1], (127)

so that

x3δ = ξ−3/2 (128)

for all ξ. The transition between the two ranges in (126) and (127) occurs at

ξ = ξT ∼ logG, (129)

since ξ increases logarithmically with r by (125). With these choices of x and δ, (120)and (121) become

F ′ = Λ min[(r/G)2/3ξ−3/2, ξ−1/2], (130)

E = (Λ3/NN ′)(1 + z)(sinh2 η)Ec ξ−3/2. (131)

Now consider the total number of bits received at B up to some epoch ξ in the remotefuture. According to (130), this number is approximately

FT =∫ ξ

F ′ dξ = 2Λξ1/2, (132)

and increases without limit as ξ increases. On the other hand, the total energy expendedby the transmitter over the entire future is nite,

ET =∫ ∞

E dξ = 2(Λ3/NN ′)(eη sinh2 η) ξ−1/2p Ec. (133)

In (133) I have replaced the red shift (1+ z) by its asymptotic value eη as ξ →∞. I havethus reached the same optimistic conclusion concerning communication as I reached inthe previous lecture about bilogical survival. It is in principle possible to communicateforever with a remote society in an expanding universe, using a nite expenditure ofenergy.

23

It is interesting to make some crude numerical estimates of the magnitudes of FT andET . By (107), the cumulative bit count in every communication channel is the same, ofthe order

FT = 1029ξ1/2, (134)

a quantity of information amply sucient to encompass the history of a complex civilization.To estimate ET , I suppose that the transmitter and the receiver each contain 1 kg ofelectrons, so that

N = N ′ = 1030. (135)

Then (133) with (105) gives

ET = 1023(eη sinh2 η) erg. (136)

This is of the order of 109 W yr, an extremely small quantity of energy by astronomicalstandards. A society which has available to it the energy resources of a solar-type star(about 1036 W yr) could easily provide the energy to power permanent communicationchannels with all the 1022 stars that lie within the sphere η < 1. That is to say, allsocieties within a red shift

z = e− 1 = 1.718 (137)

of one another could remain in permanent communication. On the other hand, directcommunication between two societies with large separation would be prohibitively expensive.Because of the rapid exponential growth of ET with η, the upper limit to the range ofpossible direct communication lies at about η = 10.

It is easier to transmit information to larger distances than η = 10 without greatexpenditure of energy, if several societies en route serve as relay stations, receiving andamplifying and retransmitting the signal in turn. In this way messages could be deliveredover arbitrarily great distances across the universe. Every society in the universe couldultimately be brought into contact with every other society.

As I remarked in the rst lecture [see Eq. (11)], the number of galaxies that lie withina sphere η < ψ grows like e2ψ when ψ is large. So, when we try to establish linkagesbetween distant societies, there will be a severe problem of selection. There are too manygalaxies at large distances. To which of them should we listen? To which of them shouldwe relay messages? The more perfect our technical means of communication become, themore diculty we shall have in deciding which communications to ignore.

In conclusion, I would like to emphasize that I have not given any denitive proof ofmy statement that communication of an innite quantity of information at a nite cost inenergy is possible. To give a denitive proof, I would have to design in detail a transmitterand a receiver and demonstrate that they can do what I claim. I have not even tried todesign the hardware for my communications system. All I have done is to show that asystem performing according to my specications is not in obvious contradiction withthe known laws of physics and information theory.

The universe that I have explored in a preliminary way in these lectures is verydierent from the universe which Steven Weinberg had in mind when he said, "Themore the universe seems comprehensible, the more it also seems pointless."I have founda universe growing without limit in richness and complexity, a universe of life survivingforever and making itself known to its neighbors across unimaginable gulfs of space andtime. Is Weinberg's universe or mine closer to the truth? One day, before long, we shouldknow.

Whether the details of my calculations turn out to be correct or not, I think I haveshown that there are good scientic reasons for taking seriously the possibility that lifeand intelligence can succeed in molding this universe of ours to their own purposes. As

24

Haldane (1924) the biologist wrote fty years ago, "The human intellect is feeble, andthere are times when it does not assert the innity of its claims. But even then:

Though in black jest it bows and nods,

I know it is roaring at the gods,

Waiting the last eclipse."

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